Journal of Hydrology 411 (2011) 279–289

Contents lists available at SciVerse ScienceDirect

Journal of Hydrology

journal homepage: www.elsevier .com/ locate / jhydrol

Can a simple stochastic model generate rich patterns of rainfall events?

Simon-Michael Papalexiou a, Demetris Koutsoyiannis a, Alberto Montanari b,⇑a Department of Water Resources, Faculty of Civil Engineering, National Technical University of Athens, Heroon Polytechneiou 5, GR-157 80 Zographou, Greeceb Department DICAM, University of Bologna, Italy

a r t i c l e i n f o

Article history:Received 22 December 2010Received in revised form 14 June 2011Accepted 10 October 2011Available online 18 October 2011This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of Uwe Haberlandt, AssociateEditor

Keywords:Long-term persistencePower-type tailsRainfall modelling

0022-1694/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jhydrol.2011.10.008

⇑ Corresponding author.E-mail address: alberto.montanari@unibo.it (A. Mo

s u m m a r y

Several of the existing rainfall models involve diverse assumptions, a variety of uncertain parameters,complicated mechanistic structures, use of different model schemes for different time scales, and possi-bly classifications of rainfall patterns into different types. However, the parsimony of a model is recog-nized as an important desideratum as it improves its comprehensiveness, its applicability and possiblyits predictive capacity. To investigate the question if a single and simple stochastic model can generatea plethora of temporal rainfall patterns, as well as to detect the major characteristics of such a model(if it exists), a data set with very fine timescale rainfall is used. This is the well-known data set of the Uni-versity of Iowa comprising measurements of seven storm events at a temporal resolution of 5–10 s. Eventhough only seven such events have been observed, their diversity can help investigate these issues. Anevident characteristic resulting from the stochastic analysis of the events is the scaling behaviors both instate and in time. Utilizing these behaviors, a stochastic model is constructed which can represent allrainfall events and all rich patterns, thus suggesting a positive reply to the above question. In addition,it seems that the most important characteristics of such a model are a power-type distribution tailand an asymptotic power-type autocorrelation function. Both power-type distribution tails and autocor-relation functions can be viewed as properties enhancing randomness and uncertainty, or entropy.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction and motivation

Rainfall has been traditionally regarded as a random processwith several peculiarities, mostly related to intermittency andnon-Gaussian behavior. However, many have been not satisfiedwith the idea of a pure probabilistic or stochastic description ofrainfall and favored a deterministic modelling option. For example,Eagleson (1970, p. 184) states ‘‘The spacing and sizing of individualevents in the sequence is probabilistic, while the internal structureof a given storm may be largely deterministic’’. Such a perceptionof rainfall is also reflected in common engineering practices, suchas the construction of design storms, in which the total depthmay be determined by probabilistic considerations but thearrangement of rainfall depth increments follows a deterministicprocedure, e.g. a pre-specified dimensionless hyetograph.

More recently, developments of nonlinear dynamical systemsand chaos allowed many to apply algorithms from these disciplinesin rainfall and claim for having discovered low dimensional deter-ministic dynamics in rainfall (see e.g., Sivakumar, 2000; Puente andSivakumar, 2007). However, such results have been disputed byothers (e.g., Schertzer et al., 2002; Koutsoyiannis, 2006a). In thelatter study, among other data sets, a high temporal resolution data

ll rights reserved.

ntanari).

record was used, in which the application of chaos detection algo-rithms did not give any indication of low dimensional chaos.

This high resolution record is one of seven storms that weremeasured by the Hydrometeorology Laboratory at the Universityof Iowa using devices that are capable of high sampling rates, onceevery 5 or 10 s (Georgakakos et al., 1994). This unique dataset allows inspection of the rainfall process at very fine time scalesand was the subject of several extensive analyses includingmultifractal analysis and multiplicative cascades (Cârsteanu andFoufoula-Georgiou, 1996) and wavelet analysis (Kumar and Foufo-ula-Georgiou, 1997). However, apart from such more technicalanalyses, this unique data set offers a basis for simpler yet morefundamental investigations that could provide insights for thecharacterization and mathematical modelling of the rainfall pro-cess; this will be attempted in the next sections. In this respect,the Iowa data set allows revisiting and acquiring better insighton the questions whether a single model can or cannot generatedifferent types of events with enormous differences among themand, if yes, how such a model would look like. First, will it bedeterministic or stochastic? A deterministic perception of therainfall process may seem in accord to the high temporal depen-dence (autocorrelation) of the rainfall process at small lag times.However, this may indicate a misconception because au fondhigh autocorrelation without a specified underlying reason (an apriori known deterministic control) may increase rather thanreduce uncertainty (Koutsoyiannis, 2010) and thus may require a

280 S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289

stochastic description. In the latter case, fundamental behaviors tobe explored are (a) the long (e.g., power-law) or short (e.g., expo-nential) tails in probability distribution function and (b) the longor short tails of the autocorrelation function. In both cases, longtails imply high uncertainty and may comply with the maximumentropy principle applied with certain constraints (Koutsoyiannis,2005a,b).

It should be emphasized from the beginning that the scope ofthis paper is more explanatory than descriptive. In this respect,some general properties of a candidate rainfall modelling ap-proach, rather than the construction of a complete and accuratemodel, are sought. Besides, as the empirical basis of this study isthe Iowa data set which comprises only seven uninterrupted singlestorms, it is impossible to study all aspects of the rainfall process.and generalize the validity of our findings for other seasons orother locations. For example intermittency, a very important pecu-liarity of the rainfall process is left out of this study. For the latter,and especially its relationship to the maximum entropy principle,the interested reader is referred to a recent study by Koutsoyiannis(2006b).

2. General properties of rainfall data set

2.1. The data

Seven storm events of high temporal resolution, recorded by theHydrometeorology Laboratory at the Iowa University (Georgakakoset al., 1994), are the data set of this study. The original measure-ments were taken every 5 or 10 s; however, for uniformity herewe use the 10-s resolution for all events. Fig. 1 illustrates the pat-terns of the seven storms.

The events are characterized by a variable duration and also ex-hibit large statistical differences among them. Specifically, sum-mary statistics like the mean, the standard deviation, theskewness and the kurtosis, shown in Table 1, differ notoriouslyamong the events, up to two orders of magnitude (e.g., the kurtosiscoefficient). In the following analyses, the different events are ana-lyzed either separately or jointly. For the latter type of analysis,which is consistent with the scope of the paper to seek whethera single model can or cannot generate all different types, a mergedsample of all events is used.

Fig. 1. The seven storm events recorded by the Hydr

2.2. Scaling in state

The term scaling in state (Koutsoyiannis, 2005a) refers to thepower-law behavior of the probability distribution of a process.Whether or not a natural process is characterized by a power-law distribution is of great importance, as a power-law processimplies that extreme events are not only more frequent in compar-ison to an exponential-law process, but also more severe. Clearly,the frequency and the magnitude of extreme events in natural pro-cesses like rainfall, have many practical consequences, e.g., in thedesign of hydraulic works.

In practice, the identification and the characterization of a nat-ural process as a power-law process is a difficult task. Natural pro-cesses that are considered to be power-law, do not exhibit a singlepower law distribution over the entire domain. Thus, the rangeover which the power-law holds, i.e. the distribution tail, mustbe identified and this is not trivial. Actually, inferences related todistribution tail that are based on sample data are uncertain.Therefore, in the best case, the validity of a power law might beconjectured, if the empirical data are consistent with the hypothe-sized power law and do not falsify the power-law hypothesis.

Generally, there are several methods for identifying power-lawbehavior in empirical data, e.g., methods based on least-square fit-ting or maximum likelihood, but none of them seems to be univer-sally accepted (see e.g., Clauset et al., 2009 and references therein).Nevertheless, one of the most common procedures used for dis-cerning power-law behavior in empirical data, which dates backto the end of 19th century in the works of Pareto, is based onleast-square fitting.

Mathematically, a random variable X follows a power-law dis-tribution, if its probability density function is of the form

fXðxÞ � LðxÞx�a�1 ð1Þ

where a > 0 is a constant known as the scaling exponent or the tailindex, and L(x) is a slowly varying function, that is a function satis-fying limx!1LðcxÞ=LðxÞ ¼ 1, where c is a constant. The essence of aslowly varying function is that asymptotically it does not affectthe power-law behavior of the distribution, thus controlling theshape of the distribution only over a finite domain of values.Straightforwardly from (1), the qth moment of a power-law distri-bution, defined as mq :¼

R1�1 xqfXðxÞdx, diverges if q > a.

ometeorology Laboratory at the Iowa University.

Table 1Summary statistics of the seven storm events.

Event # 1 2 3 4 5 6 7 All

Sample size 9697 4379 4211 3539 3345 3331 1034 29,536Mean (mm/h) 3.89 0.50 0.38 1.14 3.03 2.74 2.70 2.29Standard deviation (mm/h) 6.16 0.97 0.55 1.19 3.39 2.20 2.00 4.11Skewness 4.84 9.23 5.01 2.07 3.95 1.47 0.52 6.54Kurtosis 47.12 110.24 37.38 5.52 27.34 2.91 �0.59 91Hurst Exponent 0.94 0.79 0.89 0.94 0.89 0.87 0.97 0.89

S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289 281

It is also apparent from (1) that in a double-logarithmic plot, apower-law distribution (for both the probability density functionand the probability distribution function) would be depicted as astraight line—at least in the range of values where the power lawholds, i.e. the distribution tail. Thus, the slope of the least-square-fitted line to the tail of the empirical distribution (which,by virtue of (1) is proportional to x�a) is an estimate of the state-scaling exponent. Using the aforementioned Pareto’s method, wedepict in Fig. 2 the empirical probability distribution (constructedby using the Weibull plotting position) of the merged Iowa datasetand the least-square-fitted line to the empirical tail. A power lawwith a � 3 seems to describe the tail (at probability of exceedencesmaller than 1%).

2.3. Scaling in time

Since Hurst (1951) empirically discovered scaling in time(Koutsoyiannis, 2005b), else known as long-term persistence(LTP), this same behavior has been identified in many other naturalprocesses, as well as time series from many other scientific disci-plines, e.g., in economy and in network traffic (e.g., Baillie, 1996;Leland et al., 2002). Ever since, LTP has been an active researchfield, as its importance necessitated not only theoretical accounts,but also, practical approaches concerning primarily the estimationof its strength and the development of models capable of generat-ing synthetic time series with LTP behavior.

Basically, scaling in time can be defined in terms of the averagedprocess on several time scales k, i.e.

XðkÞðsÞ :¼ 1k

Xkst¼ðs�1Þkþ1

XðtÞ ð2Þ

In a scaling process the following expression holds, i.e.,

½XðkÞðsÞ � lX � �d kH�1½XðtÞ � lX � ð3Þ

Fig. 2. Empirical probability distribution (Weibull plotting position) of the mergedIowa dataset and the least-square-fitted line to the empirical tail.

for any t and s, where H is the scaling exponent or the so-calledHurst coefficient, and �d stands for equality in probability distribu-tion. This process has recently been termed the Hurst-Kolmogorovprocess (HK; to give credit to Kolmogorov, 1940, who was the firstto propose it). If X is Gaussian the process is also called fractionalGaussian noise (fGn), due to Mandelbrot and Van Ness (1968). Ascan be easily derived by (3), rXðkÞ ¼ kH�1rX , that is, the aggregatedprocess’s standard deviation is proportional to kH�1 and not tok�0:5 as is in the case of independent processes. In addition, theautocorrelation function qðsÞ � s2H�2 as s!1 and the spectraldensity S(x) �x1 – 2H. While in the HK process the property (3)holds for all time scales, in other processes it may hold only asymp-totically, as scale tends to infinity. Again the Hurst coefficient H isan important characteristic of the asymptotic behavior. For exam-ple, in a Markovian process, H = 0.5 (as in independent processes).

With reference to LTP identification and parameter estimation—a non-trivial issue—many methods have been developed (e.g.based on maximum likelihood, the periodogram, the variance,the rescaled range and others concepts (e.g., Taqqu et al., 1995;Taqqu and Teverovsky, 1998; Tyralis and Koutsoyiannis, 2011),each having its advantages and drawbacks.

In this study, we chose to estimate the Hurst coefficients H ofeach of the seven storm events, and additionally of the mergeddataset, by using a method that is based on the scaling propertyof the standard deviation, i.e., rXðkÞ ¼ kH�1rX . Taking the logarithms,it follows that lnrXðkÞ ¼ ðH � 1Þ ln kþ ln rX , and consequently, theaggregated sample standard deviation rXðkÞ versus the timescale kin a double-logarithmic plot, would be depicted as a straight line(at least in the timescale range where the scaling holds) and theestimated Hurst coefficient is H = 1 + g, where g is the slope ofthe fitted linear regression line.

The estimated Hurst coefficients of the seven storm events arepresented in Table 1; the variation among the estimated coeffi-cients is high, from 0.77 to 0.97 with a mean value 0.88. Neverthe-less, under the assumption that the seven storm events can beconsidered as the realizations of a single process, a better estimateof the Hurst coefficient would result if the estimation is carried outon the merged dataset, taking care in the aggregation procedurethat individual storm events do not interfere with each other. AsFig. 3 reveals, the scaling in the merged event seems to hold overthe whole range of timescales, while the estimated Hurst coeffi-cient is 0.94.

3. Stochastic analysis of the rainfall data set

3.1. The simulation scheme

As previously mentioned, the major target of this study is firstto explore if the seven storm events could be considered as the out-come of a sole and simple stochastic process, and second, to iden-tify the basic characteristics of this process. To this aim weproceeded heuristically; that is, we set a stochastic simulationscheme to generate synthetic rainfall series whose statistics aresubsequently compared with those of the observed records. Theaim is to check whether one cannot reject the hypothesis thatthe statistics themselves are coincident and therefore the observed

Fig. 3. Double logarithmic plot of sample standard deviation versus scale ofaveraging for the normalized merged event.

282 S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289

and synthetic records could be regarded as realizations of the samestochastic process. We considered different stochastic processes. Inprinciple, candidate processes should include power law as wellexponential solutions for both the marginal probability distribu-tion and the autocorrelation. As mentioned above, there are twomajor questions that need to be answered: the first, concerns thescaling in state, i.e., whether or not, the stochastic process’s mar-ginal probability distribution is power type or exponential type.The second, concerns the scaling in time, i.e., whether of not, theautocorrelation structure is power type or exponential.

Regarding the marginal distribution, it is straightforward thatrealizations from a stochastic process with a power-type marginaldistribution would exhibit large differences from an exponentialmarginal distribution, mainly because a power-type distributionassigns large probabilities to the extreme events, which signifieshigh variability and uncertainty. Clearly, this behavior is in agree-ment with the large variability observed in the seven recordedstorm events, and in addition, the whole dataset does not falsifythe power-law hypothesis of the marginal distribution (see Sec-tion 2.2). Consequently, the power-law hypothesis of the marginaldistribution is accepted as rational and valid and only stochasticprocesses with power law marginal distribution were consideredfor the simulation.

In contrast to the choice of the marginal distribution, the a pri-ori decision of a particular autocorrelation structure for the sto-chastic process is not simple. Short term persistence (STP)models have been a frequent choice in simulating natural phenom-ena, but they are often unjustifiably adopted (Koutsoyiannis andMontanari, 2007; Koutsoyiannis et al., 2009). In fact, an LTP auto-correlation structure, in many cases, may be more appropriate(see, for instance, Mandelbrot and Van Ness, 1968; Mandelbrotand Wallis, 1969). Additionally, it is not clear, how intensivelythe autocorrelation structure of a stochastic process—taking intoaccount that the marginal distribution remains the same—affectsthe variability of the sample statistics among different realizations,e.g., the statistics among the simulated storm events addressed inthis study. Thus, even in the case when the empirical evidence sup-ports the adoption of a certain autocorrelation structure, and inview of the intrinsic uncertainty of this choice, it is valuable to per-form a comparison of different scenarios, i.e., a comparison be-tween STP and LTP autocorrelation structures. Therefore, thisrationale suggests a side-by-side comparison between an STP mod-el and an LTP model in view of the behaviors of the observed data.

The following sections present the simulation scheme whichconsists of the following seven steps: (1) application of an

appropriate normalizing transformation to the original dataset(Section 3.2); (2) analysis of the empirical ACF (Section 3.4); (3)identification and calibration of an STP model and an LTP model(Sections 3.5 and 3.6) to the normalized dataset; (4) correction ofthe model standard deviation bias (Section 3.7); (5) simulation ofnormal synthetic time series (Section 3.8); (6) generation of thesynthetic rainfall time series by applying the inverse transforma-tion (see Section 3.2) to the normal synthetic time series; and (7)statistical analysis of the synthetic time series (Section 4).

3.2. Normalizing the original data

The Gaussian or the Normal distribution is probably the mostknown and the most widely used distribution in statistics, withapplications also in natural sciences. There are two theoretical rea-sons that justify the ubiquity of the Normal distribution in statis-tics and its application in other scientific fields. The first, relatesto the central limit theorem (CLT) that—loosely speaking—statesthat the sum of independently and identically distributed (i.i.d)random variables tends to the Normal distribution as the numberof summands tends to infinity. The second is the principle of max-imum entropy (Jaynes, 1957), which states that, among all possibledistributions with known mean and variance, the normal distribu-tion is the one that maximizes the Boltzmann–Gibbs–Shannoninformation entropy (see also Shannon Claude and Weaver, 1948).

Nevertheless, it seems that geophysical data are seldom normal.Empirical data show that many geophysical processes, like rainfalland river discharge, may depart mildly or severely from normality,especially at small time scales. A relevant example is the datasetaddressed in this study. Specifically, departures from normalitymay be identified in skewness, e.g., positively or negatively skewedempirical data, in the asymptotic behavior of the distribution tail,e.g., a stretched exponential tail or a power-type tail, and of course,in the variable’s domain. Thus, as there exist theoretical reasonsthat favor normality in many cases, theoretical reasons also existthat do not support it (see e.g., Koutsoyiannis, 2005a).

For instance, it is well known that a normal variable ranges overthe whole real axis, while many natural processes are positivelydefined, that is, have a lower limit at zero, while a solid reasonto fix an upper limit very rarely exists. While the previous reasonsexplain why departures from normality are so common in nature, aformal and generalized method for simulating non-normal datawith a certain autocorrelation structure does not exist, althoughheuristic solutions were frequently proposed (for a hydrologicalexample, see Montanari et al., 1997). In contrast, several methodsexist addressing the simulation of normal data with STP or LTPautocorrelation structures (e.g., Box et al., 1994; Koutsoyiannis,2000; Brockwell and Davis, 2009). A common technique for simu-lating non-normal data consists of transforming the non-normaldataset to normal, by applying a normalizing transformation, next,simulating normal data by implementing a standard model, and fi-nally, de-normalizing the normal data by applying the inversetransformation. Basically, this is the methodology followed alsoin this study, which presents the inconvenience that finding anappropriate normalizing transformation is not always a trivial task,and clearly, a general method for normalizing all types of data doesnot exist. It is well known that there are some general and com-monly used families of transformations, like the Box-Cox familyof transformations (Box and Cox, 1964), that in many cases givesatisfactory results. Unfortunately, such general and simple trans-formations were not effective for the case of the Iowa dataset. Inparticular, while the application of the Box-Cox transformation re-sulted in approximately normal data for the upper empirical tail, itfailed to normalize the lower tail, namely the values near zero. Afrequently used solution to solve this problem is the normal quan-tile transform (also called normal quantile score; Kelly and

Fig. 4. Probability plot of the natural (recorded) and the normalized rainfallintensity data.

S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289 283

Krzysztofowicz, 1997) which, however, is an empirical transforma-tion that is defined over the range of the observed data only andcannot be extrapolated.

Therefore, in order to normalize the Iowa dataset we introducehere heuristically (by extending a transformation by Koutsoyianniset al., 2008) a five-parameter normalizing transformation given by

zðtÞ ¼ gðxðtÞÞ

¼ ½axðtÞ�f þ b� cþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

d

� �lnf1þ d½xðtÞ � c�2g

s !ð4Þ

where z(t) and x(t) are the transformed and original values of therainfall intensity, which are realisations of the stochastic processesZ(t) and X(t), respectively, and a, b, c, d, f are the parameters to beestimated. The two factors of the product in the right hand sideare introduced to normalize the lower and the larger values,respectively.

While this transformation was identified heuristically, itsconstruction was based on two theoretical aspects. First, equation(4) ensures that the random variable Z � N(0,1) ranges from �1to 1. Obviously, inspection of (4) reveals that for fa; b; d; fg 2ð0;1Þ and c 2 ð�1;0Þ, the random variable Z 2 ð�1;1Þ, aslimx!0þgðxÞ ¼ �1 and limx!1gðxÞ ¼ 1. Second, the probabilitydensity function (pdf) of the random variable X should be longtailed as the empirical evidence supports this assumption (seeSection 2.2). Again, inspection of (4) reveals that for large valuesof x, g(x) � [2b2 (1 + 1/d)ln x]0.5, and taking into account thatfZ(z) � exp(z–2/2) and combining the two equations, we getfX(x) � fZ(g(x)) � x�b2 (1+1/d) and thus the pdf of the variable X is longtailed.

Finally, we estimated the parameters of (4) for the transformedmerged Iowa dataset by using the method of least-squares, andparticularly, we numerically minimized the sum of squared errorsbetween values of the standardized normal variate that correspondto the values of the empirical normal distribution (obtained byapplying the normal quantile transformation) and the respectivevalues result from the application of (4) to the original rainfall val-ues. The resulted estimates were a = 0.41, b = 2.49, c = �2.13,d = 4.09 and f = 1.18. The transformed data in comparison withthe original data are presented in Fig. 4. Clearly, as the Fig. 4 dem-onstrates, the transformed data are satisfactorily normalized.

3.3. Identification and calibration of the stochastic models

A Gaussian (normal) stochastic process is completely character-ized when its second-order distribution, i.e., FXðxi; xj; ti; tjÞ ¼PfXðtiÞ 6 xi;XðtjÞ 6 xjg for any i – j, is known. Normalizing themarginal distribution of a stochastic process by a transformation,does not necessarily result in jointly normal distribution (Feller,1971, p.70). However, it is important to check if a particular, mar-ginally normalized, data has also become Gaussian in terms of themultivariate joint distribution or not. A rough indication of jointnormality is provided by the linear relation of conditional expecta-tion of a variable Xi given Xj for i – j. Fig. 5 depicts the normalizedrainfall intensity versus the 1-time-step and 10-time-step shiftednormalized rainfall intensity. It can be seen that the empiricalpoints are spread around a straight line, which is an indication ofjoint normality. This linearity should not be regarded as a surprise,given that it is consistent with the principle of maximum entropyapplied on a multivariate setting with constraints of known mean,variance and lag-1 autocorrelation.

As discussed in Section 2.3, scaling in time exists and is quanti-fied by an estimated Hurst coefficient H = 0.94. Similarly, analysis ofthe transformed dataset, using the same methods as in Section 2.3,reveals also a high value of the Hurst coefficient, i.e., H = 0.92. Thus,

if we accept the assumption of scaling in time, a serious issue arises;that is, almost all classical estimators of statistics (exception is themean value) are highly biased (e.g., Koutsoyiannis, 2003). So in or-der to set up an accurate and consistent stochastic model to simu-late a normal process—that is, a model that sufficiently reproducesthe mean, the standard deviation and the autocorrelation structureof the observed sample—unbiased and accurate estimates of theaforementioned statistics are necessary.

3.4. Empirical autocorrelation function (ACF)

It is well known, that for finite samples the typical estimate q̂l

of the lag-l autocorrelation is a biased estimator of the true auto-correlation ql and the more intense the autocorrelation structureis the more biased the estimator becomes. In particular, in thepresence of scaling in time the bias can be corrected by the follow-ing formula (see Koutsoyiannis, 2003 and the references therein),

~ql ¼ q̂l 1� 1n2�2H

� �þ 1

n2�2Hð5Þ

where ~ql stands for the unbiased estimator and H is the Hurstcoefficient.

In this study, we used the unbiased estimator in (5) to estimatethe empirical autocorrelation coefficients. We clarify, that to esti-mate q̂l and consequently to estimate the unbiased estimator givenin (5), we used the transformed merged sample that comprises theseven transformed storm events. This is a reasonable choice if wereckon the seven events as the outcome of a single process; andthus, while the empirical ACF may differ among events, all eventsshare the same theoretical ACF. Furthermore, we note that we tookspecial care in the estimation of the covariance in order to avoidoverlapping among the events; specifically, we eliminated all prod-ucts of the form ðxt � l̂XÞðxt�l � l̂XÞ when xt and xt�l do not belongin the same storm event, and adjusted accordingly the number n ofthe sample size. The estimated unbiased empirical ACF—given aHurst coefficient equal to H = 0.92, and for lags approximately upto 1000—is depicted in Fig. 6. Clearly, as Fig. 6 attests, the empiricalautocorrelation structure is very intense, and particularly, the val-ues of the small-lag autocorrelation coefficients are near to 1, whilefor lags near to 1000 the values are as high as 0.85.

3.5. The short-term persistence model

Probably, the most common STP stochastic model is the lag-oneautoregressive model AR(1). This model belongs to the general

Fig. 5. Scatter plot of normalized rainfall intensity for time lags 1 and 10.

284 S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289

family of stochastic models known as autoregressive moving-aver-age models ARMA(p,q)—comprehensively presented in Box et al.(1994). It is important to note that the ARMA(p,q) family, and espe-cially the AR(1) model are not able to reproduce the scaling behav-ior in time or to preserve the Hurst coefficient (e.g., Koutsoyiannis,2002). Consequently, they may be inappropriate for simulatingnatural phenomena exhibiting LTP.

Nevertheless, while from a theoretical viewpoint ARMA(p,q)models are considered STP models, for increasing values of theautoregressive and moving average order p and q they can providevery good approximations of the LTP structure and thus manage toreproduce, from a practical point of view, the scaling in time or topreserve the Hurst coefficient at least for small sample sizes(Papalexiou, 2007). It is clear, though, that high order ARMA(p,q)models are not parsimonious, i.e., many parameters need to beestimated therefore increasing the estimation variance.

Here, we chose the ARMA(2,2) model for the simulation of thenormalized rainfall intensity. It is a model frequently used inhydrology that is able to generate time series that preserve themean value lX, the variance r2

X and the first four autocorrelationcoefficients fq1;q2;q3;q4g. The stochastic process fXðtÞ; t 2 Tg thatresults from an ARMA(2,2) model is defined by

XðtÞ ¼ a1Xðt � 1Þ þ a2Xðt � 2Þ þ b1eðt � 1Þ þ b2eðt � 2Þ þ eðtÞ ð6Þ

Fig. 6. Empirical ACF of the normalized merged event (corrected for bias),theoretical ACF of the fitted STP model, fitted power-type ACF given in eq. (9),and approximation of the latter by the sum of five AR(1) processes.

where fa1; a2;b1;b2g are parameters, and eðtÞ is a normal white-noise process, i.e. consisting of independently, identically and nor-mally-distributed random variables with mean le ¼ 0 and variancer2

e . Using typical estimation methods (Box et al., 1994), the result-ing parameters for the transformed merged Iowa dateset arefa1 ¼ 1:51; a2 ¼ 0:51;b1 ¼ �0:57;b2 ¼ �0:19;re ¼ 0:11g.

Once the model parameters are estimated, the theoretical ACFof the ARMA(2,2) for lags s P 3 degenerates to the ACF of anAR(2), i.e., qs ¼ a1qs�1 þ a2qs�2 and thus can be calculated recur-sively. Fig. 6 depicts the theoretical ACF of the fitted ARMA(2,2)model in comparison with the empirical ACF. Clearly, it preservesthe first four autocorrelation coefficients, as expected, and also per-forms well for lags up to 50. Nevertheless, for higher lags, it clearlydeviates from the empirical ACF as the exponential character of thetheoretical ACF unfolds.

3.6. The long-term persistence model

Since the time when Hurst (1951) discovered the LTP behavior,the necessity to consistently simulate natural phenomena that ex-hibit LPT has led to the development of several stochastic processesand algorithmic procedures that reproduce the LTP behavior.Among the most common models are several algorithmic approx-imations of the HK (or fGn) process by Mandelbrot and Wallis(1969) Mandelbrot (1971), O’Connell (1974), Koutsoyiannis(2002), and the FARIMA(p,d,q) models introduced by Granger andJoyeux (1980) and Hosking (1981), that have gained popularitymainly in the last decade (for an application to hydrology seeMontanari et al., 1997).

The theoretical ACFs of the HK and FARIMA(0,d,0) processes are

qFGNs ¼ 1

2ðjs� 1j2H � 2jsj2H þ jsþ 1j2HÞ � s2H�2 ð7Þ

qFARIMAs ¼ Cð1� dÞCðsþ dÞ

CðdÞCðsþ 1� dÞ � s2d�1 ð8Þ

respectively. Clearly, the ACFs of those two models are asymptoti-cally coincident, with d = H � 1/2, as (7) and (8) attest, whereas,time series generated by both of them preserve the scaling expo-nent H. Moreover, while the HK process model is a very simplemodel—essentially is one-parameter model, the FARIMA(p,d,q)models are much more flexible as the orders of p and q controlsthe STP behavior of the model.

Fig. 7. Standard deviation bias correction factors for the STP and the LTP models forvarious sample sizes (dots and triangles) calculated by Monte Carlo simulation.

S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289 285

Here we used a simple yet general approach to simulate LTP,obtained by approximating the real process with the sum of fiveindependent AR(1) processes (note that Koutsoyiannis (2002) hasshown that good approximations can be obtained even with sum-ming three independent AR(1) processes). The implementationcomprises two steps: first; we fit a generalized power-type (GP)ACF to the empirical ACF (see 3.4) and second; we approximatethe fitted ACF by the ACF obtained as the sum of five independentAR(1).

Regarding the first step of our approach, we fit a theoretical ACF(consistent with the empirical evidence) to the empirical ACF in or-der to be able to extrapolate the correlation coefficients for lags ashigh as desired, instead of being confined in the lag-range providedby the estimated empirical ACF.

Here, we use a theoretical three-parameter power-type ACFthat has the form (similar to Gneiting and Schlather, 2004)

qGPs :¼ ½1þ cðs

aÞb��

1c ð9Þ

where fa > 0; b > 0; c > 0g are parameters. The form of (9) can beconsidered as a natural generalization of an exponential ACF asthe limc!0qGP

s ¼ expð�s=aÞb. Asymptotically (9) behaves as qGPs �

t –b/c and therefore, (9) and (7) possesses the same asymptoticbehavior if b=c ¼ 2ð1� HÞ. As a result, the fitted qGP

s would be con-sistent with the estimated H = 0.92 if b/c = 0.16. Thus, we fit the qGP

sby minimizing the square error between the qGP

s and the empiricalACF and by setting as a constrain b/c = 0.16. The estimated param-eters are fa ¼ 12881; b ¼ 0:51; c ¼ 3:18g. The fitted qGP

s is depictedin Fig. 6, which shows that the fit is satisfactory.

Turning to the second step of the LTP simulation procedurementioned above, we decided to use an LTP model made up bythe sum of five independent AR(1) process by following the ideathat was first introduced by Mandelbrot (1971), to approximatethe HK process. The same method was used by Koutsoyiannis(2002), for the same purposes, while Mudelsee (2007) provedempirically that the sum of n inflows generated by an AR(1) modelin a river network, with n sufficiently large, ends up with a collec-tive river discharge that exhibits LTP behavior.

Therefore the LTP model that was used herein to simulate thenormalized rainfall intensity is given by

YðtÞ ¼X5

i¼1

YiðtÞ ð10Þ

where YiðtÞ ¼ aiYiðt � 1Þ þ eiðtÞ is the i-th AR(1) process with meanlYi¼ 0, variance rYi

, lag-one autocorrelation coefficient ai and eiðtÞis a normal white-noise process, with mean lei

¼ ð1� aiÞlYi¼ 0

and variance r2ei¼ ð1� a2

i Þr2Yi

. Under the assumption of indepen-dence of the five AR(1) processes it can be easily proven that thetheoretical ACF of (10) is given by

qLTPs ¼

X5

i¼1

asi r

2Yi; with

X5

i¼1

r2Yi¼ 1 ð11Þ

The parameters of the five independent AR(1) processes wereestimated by minimizing the square error between Eqs. (9) and(11) for lags as high as 104. The resulting estimates are fa1 ¼0:9943;r2

Y1¼ 0:075g, fa2 ¼ 0:8719;r2

Y2¼ 0:029g, fa3 ¼ 0:999932;

r2Y3¼ 0:179g, fa4 ¼ 0:9994;r2

Y4¼ 0:138g and fa5 ¼ 0:999998;

r2Y5¼ 0:578g. As the Fig. 6 reveals, the fitted qLTP

s , up to the lag-104, is satisfactory.

3.7. The standard deviation bias

One issue in stochastic modelling that may have serious conse-quences on the validity and accuracy of the simulation, and is often

neglected, concerns the differences in statistics that may occur be-tween the theoretical process and its realizations. While the esti-mate of the mean is unbiased regardless of the dependencestructure, this does not hold for the standard deviation. In fact, itis well known that the standard estimator S of the standard devia-tion is slightly biased even in the case of normally distributed andindependent data (e.g., Bolch, 1968). However, the bias may be-come very large in a time dependent process as it increases mono-tonically with the increase of the autocorrelation. For certainknown ACFs, like the one of the HK process, unbiased estimatorshave been developed (see Koutsoyiannis, 2003 and referencestherein).

In order to assess the standard deviation bias in random sam-ples generated by the STP and the LTP models described in Sections3.5 and 3.6, and for several different sample sizes, we performed aMonte Carlo simulation. Specifically, at first, 5000 independentsamples were generated by each model and for several samplesizes, and in turn, a standard deviation correction factor was calcu-lated, defined by cSD :¼ r=E½S� where r is the true standard devia-tion of the STP and the LTP models, chosen as 1 in our simulationsand E[S] is calculated by Monte Carlo simulation.

The results, that are depicted in Fig. 7, are remarkable, espe-cially in the case of the LTP model. In fact, the bias correction fac-tors, for a sample size of 1000, are as high as 1.9 and 3.1 for the STPand the LTP models, respectively, while even for a very large sam-ple size equal to 50,000, in the LTP case, the correction factor sus-tains a value of 1.7. Given that the correction factor depends of thesample size, the choice of the appropriate correction factor shouldbe carried out by considering the number of the data generatedwith the simulation. Given that the normalizing transformationwas applied to a sample of 29,536 values that comprised the sevenstorm events, it follows that we imposed a unit standard deviationto that complete sample. Consequently, all samples generated inthis study, irrespective of their size, were multiplied by the correc-tion factor that corresponds to a size of 29,536, that is, cSD ¼ 1:04for the STP model and cSD ¼ 1:81 for the LTP model. In this waythe correction to the standard deviation was imposed dependingon the sample size that was used to constrain the standard devia-tion itself during the normalizing transformation.

3.8. Sample size and number of samples

As shown in Table 1, the seven recorded storm events have alldifferent sample lengths varying form 1034 to 9697 values. In

Fig. 8. Synthetic rainfall events generated by the LTP (the first three) and the STP (the last three) models for three characteristic samples sizes.

Fig. 9. Box plots of sample statistics estimated from the synthetic rainfall events generated by the LTP and STP models for the three characteristic sample sizes L1, L2 and L3.The dots represent the empirical points of the seven rainfall events.

286 S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289

order to compare the observed statistics with those as the syn-thetic series, it would be appropriate that the simulations havethe same length of the observed records. For practicality only 3

sample sizes were used namely: 1000 (L1), which is very close tothe size of event 7; 4000 (L2), close to the size of events from 2to 6; and 10,000 (L3) representing event 1.

S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289 287

We generated 10,000 synthetic series for each sample lengthand for each model. Finally, we calculated for every synthetic seriesthe mean, the standard deviation, the skewness, the kurtosis andthe autocorrelations, which were compared with the respectivestatistics of the observed records.

4. Results of the stochastic simulation

Fig. 8 reports an example of visualized simulated events for thethree different sample sizes (L1, L2 and L3) considered here and thetwo different models (three events generated by the LTP model onthe left and three by the STP model on the right). Some differencesin the patterns generated by the models are visible. For example,the pattern of the LTP model is characterized by a higher variability(although the marginal distributions are the same). By comparingthe patterns with those of the observed records, which are shownin Fig. 1, one may notice that the variability of the observed recordlooks better reproduced by the LTP model.

Fig. 9 shows box plots of selected statistics computed on thesimulated data (in this case also by referring to the original prob-ability distribution), namely, mean, standard deviation, skewnessand kurtosis. The observed statistics are also shown with dots.The box and the whiskers encompass 50% and 99%, respectively,of the computed statistics, while the median is indicated by a hor-izontal straight line. The box plots clearly show the differentbehaviors of the two models. Looking at the mean value, oneshould note that, not surprisingly, the two models are character-ized by nearly the same median of the mean, but the variabilityin the LTP model is higher. Also expected is the higher variabilityof the standard deviation, skewness and kurtosis that is depictedin the other box plots. One may note that the LTP model is moreskewed than the STP one. This result is explained by the higher

Fig. 10. Empirical autocorrelation functions of the seven rainfall event

variability of a process (rainfall) that is bounded at zero. In generalone may note that the higher uncertainty of the LTP model makesthe fit more satisfactory, even though the observed points are veryfew and therefore do not allow more than a qualitative assessment.

Fig. 10 shows a comparison between the observed autocorrela-tion functions with those simulated by the models. First of all, onenotes that the autocorrelation coefficients of the LTP model arehigher in the tail of the autocorrelation function. This result is ex-pected. Another relevant feature is the higher correlations showedby the STP model for low lags. This result, which is not intuitive, isdue to the fact that the STP model, in order to reach a better fit ofthe tail of the ACF, reacts by increasing also the autocorrelationcoefficients for low lags. Conversely, the power law behavior ofthe LTP model allows one to reach a better fit of the tail of theACF without increasing much the correlation for low lags. Evenin this case, the autocorrelation function of the LTP appears to bemore convincing in view of the observed pattern. One should notethat this assessment is again qualitative in view of the small num-ber of observed events.

However, apart from the comparison between the two models,one relevant conclusion is that both models look able to provide,within a relatively simple framework, a satisfactory fit of the ob-served behaviors.

5. Conclusions and discussion

Summarizing the above investigations, it can be said that a sin-gle and rather simple stochastic model can represent all rainfallevents and all rich patterns appearing in each of the separateevents making them look very different from one another. Froma practical view point, such a model is characterized by high auto-correlation at fine scales, slowly decreasing with lag, as well as by

s and 99% prediction intervals of ACF for the LTP and STP models.

288 S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289

distribution tails slowly decreasing with rainfall intensity. Such anautocorrelation form can indeed produce huge differences amongdifferent events and such a distributional form can produce enor-mously high rainfall intensities at times. Both these behaviors arejust opposite to the more familiar processes resembling Gaussianwhite noise, which would produce very ‘‘stable’’ events with infre-quent high intensities. In this respect, both high autocorrelationsand distribution tails can be viewed as properties enhancing ran-domness and uncertainty (or entropy).

Whether the tails of both the marginal distribution and autocor-relation functions are long (meaning that are described by power-law functions) is difficult to conclude based merely on the data setof this study. Both these power-law functions are by definitionasymptotic properties, and the exponents of power laws are theo-retically defined for state or lag tending to infinity. In this respect,it seems impossible to verify such asymptotic laws by empiricalstudies, which necessarily imply finite sample sizes. But it isimportant that the empirical evidence presented in the currentstudy does not falsify the hypothesis that both tails are long. Otherempirical studies published recently do not falsify this hypothesisas well. Thus, Koutsoyiannis (2004a, 2004b) used daily rainfall re-cords from numerous gauges worldwide, each covering a periodlonger than 100 years, and showed that the observed behaviorsare consistent with the long distributional tail hypothesis.

If the hypothesis of a long tail of the distribution function is ac-cepted, it seems that this can be quantified by an exponent a ofabout 3, which implies that only the first three moments of the dis-tribution exist whereas all others are infinite. If the hypothesis of along tail of the autocorrelation function is accepted, it seems thatthis can be quantified by a Hurst coefficient H as high as 0.94.Based on these findings, the construction of a stochastic modeladmitting asymptotically long tails from the outset seems a rea-sonable choice. After all, in a dynamical systems context, eventhe randomness is an asymptotic property per se, in the sense thatit implies an infinite number of degrees of freedom. The fact thatan infinite number of degrees of freedom cannot be verified (andperhaps neither falsified) empirically, does not preclude us fromsuccessfully using probabilistic descriptions and stochastic modelsof several processes including rainfall.

As mentioned earlier, long tails can be viewed as an enhance-ment of randomness and uncertainty in these processes. In theframework of this enhanced randomness, it seems to be uselessto analyze each rainfall event separately as an attempt to inferdynamics of rainfall. Such an attempt, even using sophisticatedmethods such as wavelets, can perhaps be paralleled with one’s at-tempt to explain the dynamics of the tossing of a coin by observinga series of ‘‘heads’’ and ‘‘tails’’. In both cases, it may be misleadingto seek substantial information in extremely random occurrences.A more useful target for such cases would be to elevate from theobscurity the underlying randomness and seek its own laws –and this was the main target of this paper.

Acknowledgment

We thank Geoff Pegram and one additional anonymous re-viewer for their encouraging and constructive comments on twosubsequent versions of the paper. We thank as well one anony-mous reviewer for providing negative comments on the firstversion.

References

Baillie, R.T., 1996. Long memory processes and fractional integration ineconometrics. Journal of Econometrics 73, 5–59.

Bolch, B.W., 1968. More on unbiased estimation of the standard deviation. AmericanStatistician 22, 27.

Box, G.E., Cox, D.R., 1964. An analysis of transformations. Journal of the RoyalStatistical Society. Series B (Methodological) 211–252.

Box, G.E., Jenkins, G.M., Reinsel, G.C., 1994. Time Series Analysis: Forecasting andControl, 3rd ed. Prentice Hall, New Jersey.

Brockwell, P.J., Davis, R.A., 2009. Time Series: Theory and Methods. Springer Verlag.Cârsteanu, A., Foufoula-Georgiou, E., 1996. Assessing dependence among weights in

a multiplicative cascade model of temporal rainfall. Journal of GeophysicalResearch 101, 26363.

Clauset, A., Shalizi, C.R., Newman, M.E., 2009. Power-law distributions in empiricaldata. SIAM Review 51, 661–703.

Eagleson, P.S., 1970. Dynamic Hydrology. McGraw-Hill Inc., New York.Feller, W., 1971. An introduction to probability theory and its applications, 2nd ed.

John Wiley & Sons Inc, New York.Georgakakos, K.P., Carsteanu, A.A., Sturdevant, P.L., Cramer, J.A., 1994. Observation

and analysis of Midwestern rain rates. Journal of Applied Meteorology 33,1433–1444.

Gneiting, T., Schlather, M., 2004. Stochastic models that separate fractal dimensionand the Hurst effect. Society for Industrial and Applied Mathematics Review 46(2), 269–282.

Granger, C.W.J., Joyeux, R., 1980. An introduction to long-memory time series andfractional differencing. Journal of Time Series Analysis 1, 15–29.

Hosking, J.R., 1981. Fractional differencing. Biometrika 68, 165–176.Hurst, H.E., 1951. Long-term storage capacity of reservoirs. Transactions of the

American Society of Civil Engineers 116, 770–808.Jaynes, E.T., 1957. Information theory and statistical mechanics. II.. Physical Review

108, 171–190.Kelly, K.S., Krzysztofowicz, R., 1997. A bivariate meta-Gaussian density for use in

hydrology. Stochastic Hydrology and Hydraulics 11, 17–31.Kolmogorov, A.N., 1940. Wienersche Spiralen und Einige Andere Interessante

Kurven in Hilbertschen Raum. Doklady Akademii nauk URSS 26, 115–118.Koutsoyiannis, D., 2002. The Hurst phenomenon and fractional Gaussian noise

made easy. Hydrological Sciences Journal 47, 573–596.Koutsoyiannis, D., 2003. Climate change, the Hurst phenomenon, and hydrological

statistics. Hydrological Sciences Journal 48, 3–24.Koutsoyiannis, D., 2004a. Statistics of extremes and estimation of extreme

rainfall, 1, Theoretical investigation. Hydrological Sciences Journal 49, 575–590.

Koutsoyiannis, D., 2004b. Statistics of extremes and estimation of extreme rainfall,2, Empirical investigation of long rainfall records. Hydrological Sciences Journal49, 591–610.

Koutsoyiannis, D., 2005a. Uncertainty, entropy, scaling and hydrological stochastics.1. Marginal distributional properties of hydrological processes and statescaling.. Hydrological Sciences Journal 50, 381–404.

Koutsoyiannis, D., 2005b. Uncertainty, entropy, scaling and hydrological stochastics.2. Time dependence of hydrological processes and time scaling.. HydrologicalSciences Journal 50, 1–426.

Koutsoyiannis, D., 2006a. On the quest for chaotic attractors in hydrologicalprocesses. Hydrological Sciences Journal 51, 1065–1091.

Koutsoyiannis, D., 2006b. An entropic-stochastic representation of rainfallintermittency: the origin of clustering and persistence. Water ResourcesResearch 42.

Koutsoyiannis, D., 2010. A random walk on water. Hydrology and Earth SystemSciences 14, 585–601.

Koutsoyiannis, D., 2000. A generalized mathematical framework for stochasticsimulation and forecast of hydrologic time series. Water Resource Research 36,1519–1533.

Koutsoyiannis, D., Montanari, A., Statistical analysis of hydroclimatic time series:Uncertainty and insights, Water Resources Research, 43 (5), W05429,doi:10.1029/2006WR005592, 2007.

Koutsoyiannis, D., Yao, H., Georgakakos, A., 2008. Medium-range flow prediction forthe Nile: a comparison of stochastic and deterministic methods. HydrologicalSciences Journal 53 (1), 142–164.

Koutsoyiannis, D., Montanari, A., Lins, H.F., Cohn, T.A., 2009. Climate, hydrology andfreshwater: towards an interactive incorporation of hydrological experienceinto climate research—discussion of ‘‘The implications of projected climatechange for freshwater resources and their management’’. Hydrological SciencesJournal 54 (2), 394–405.

Kumar, P., Foufoula-Georgiou, E., 1997. Wavelet analysis for geophysicalapplications. Reviews of Geophysics 35, 385–412.

Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V., 2002. On the self-similarnature of Ethernet traffic (extended version). Networking, IEEE/ACMTransactions on 2, 1–15.

Mandelbrot, B.B., Van Ness, J.W., 1968. Fractional Brownian motions, fractionalnoises and applications. SIAM Review 10, 422–437.

Mandelbrot, B.B., 1971. A fast fractional Gaussian noise generator. Water ResourceResearch 7, 543–553.

Mandelbrot, B.B., Wallis, J.R., 1969. Computer experiments with fractional Gaussiannoises: part 1, averages and variances. Water Resource Research 5, 228.

Montanari, A., Rosso, R., Taggu, M., 1997. Fractionally differenced ARIMA modelsapplied to hydrologic time series: identification, estimation, and simulation.Water Resources Research 33, 1035–1044.

Mudelsee, M., 2007. Long memory of rivers from spatial aggregation. WaterResources Research 43, W01202.

O’Connell, P.E., 1974. A simple stochastic modelling of Hurst’s law. In: Proc.International Symposium on Mathematical Models in Hydrology, Warsaw(1971), IAHS Publ. No. 100, 169–187.

S.M. Papalexiou et al. / Journal of Hydrology 411 (2011) 279–289 289

Papalexiou, S., 2007. Stochastic modelling of skewed data exhibiting long-rangedependence. In: XXIV General Assembly of the International Union of Geodesyand Geophysics.

Puente, C.E., Sivakumar, B., 2007. Modeling geophysical complexity: a case forgeometric determinism. Hydrology and Earth System Sciences 11, 721–724.

Schertzer, D., Tchguirinskaia, I., Lovejoy, S., Hubert, P., Bendjoudi, H., Larchvêque, M.,2002. Which chaos in the rainfall–runoff process. Hydrological Science Journal47, 139–148.

Shannon Claude, E., Weaver, W., 1948. The mathematical theory of communication.Bell System Technical Journal 27, 379–423.

Sivakumar, B., 2000. Chaos theory in hydrology: important issues andinterpretations. Journal of Hydrology 227, 1–20.

Taqqu, M.S., Teverovsky, V., 1998. On estimating the intensity of long-rangedependence in finite and infinite variance time series. A Practical Guide toHeavy Tails: Statistical Techniques and Applications 177, 218.

Taqqu, M.S., Teverovsky, V., Willinger, W., 1995. Estimators for long rangedependence. Fractals 3, 785–798.

Tyralis, H., Koutsoyiannis, D., 2011. Simultaneous estimation of the parameters ofthe Hurst-Kolmogorov stochastic process. Stochastic Environmental Research &Risk Assessment 25 (1), 21–33.